Geometrical interpretation and approximate solution of non‐linear KdV equation

Abstract

Purpose

The purpose is to study an analytical solution of non‐linear Korteweg‐de Vries (KdV) equation by using the Adomian decomposition method (ADM).

Design/methodology/approach

The solution is calculated in the form of a series with easily computable components. The non‐linear KdV equation has been considered and the analytic solution is compared with its numerical solution by using the ADM and Mathematica software program.

Findings

This approach to the non‐linear evolution equation was found to be valuable as a tool for scientists and applied mathematicians, because it provides immediate and visible symbolic terms of analytical solution as well as its numerical approximate solution to both linear and non‐linear problems without linearization or discretization.

Research limitations/implications

This geometrical interpretation and the produced approximate solution of the non‐linear KdV equation illustrates the use of the ADM. Research using ADM is ongoing but already the numerical results obtained in this paper justify the advantages of this methodology, even in a few terms of approximation.

Practical implications

Using the Mathematica software package the ADM was implemented for homogenous KdV equation as an illustrative example which has distinct applications for scientists and applied mathematicians.

Originality/value

This is an original study of the use of ADM for the solution of the non‐linear KdV equation. It also shows how the Mathematica software package can be used in such studies.

Citation

Bektas, M., Inc, M. and Cherruault, Y. (2005), "Geometrical interpretation and approximate solution of non‐linear KdV equation", Kybernetes, Vol. 34 No. 7/8, pp. 941-950. https://doi.org/10.1108/03684920510605768